Inversion

Inversion is a transformation of the plane with respect to a fixed circle (called the circle of inversion) with center $O$ and radius $r$. For any point $P$ (not equal to $O$), its image $P'$ is defined so that $O$, $P$, and $P'$ are collinear, and $OP \times OP' = r^2$. The center $O$ does not have an image under inversion.

Important properties

• Points inside the circle of inversion are mapped outside, and vice versa.

• The image of a circle passing through the center of inversion is a straight line.

• The image of a line not passing through the center is a circle passing through the center.

• Inversion preserves angles between curves (it is a conformal transformation).